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Properties

Label 2-816-204.191-c1-0-23
Degree $2$
Conductor $816$
Sign $0.374 + 0.927i$
Analytic cond. $6.51579$
Root an. cond. $2.55260$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.22 + 1.22i)3-s + (0.707 + 0.707i)5-s + (1.73 − 1.73i)7-s − 2.99i·9-s + (−3.67 − 3.67i)11-s + 3·13-s − 1.73·15-s + (−3.53 − 2.12i)17-s − 5.19·19-s + 4.24i·21-s + (−1.22 − 1.22i)23-s − 4i·25-s + (3.67 + 3.67i)27-s + (2.82 + 2.82i)29-s + (−6.92 − 6.92i)31-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)3-s + (0.316 + 0.316i)5-s + (0.654 − 0.654i)7-s − 0.999i·9-s + (−1.10 − 1.10i)11-s + 0.832·13-s − 0.447·15-s + (−0.857 − 0.514i)17-s − 1.19·19-s + 0.925i·21-s + (−0.255 − 0.255i)23-s − 0.800i·25-s + (0.707 + 0.707i)27-s + (0.525 + 0.525i)29-s + (−1.24 − 1.24i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 816 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.374 + 0.927i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 816 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.374 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(816\)    =    \(2^{4} \cdot 3 \cdot 17\)
Sign: $0.374 + 0.927i$
Analytic conductor: \(6.51579\)
Root analytic conductor: \(2.55260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{816} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 816,\ (\ :1/2),\ 0.374 + 0.927i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.791914 - 0.533971i\)
\(L(\frac12)\) \(\approx\) \(0.791914 - 0.533971i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (1.22 - 1.22i)T \)
17 \( 1 + (3.53 + 2.12i)T \)
good5 \( 1 + (-0.707 - 0.707i)T + 5iT^{2} \)
7 \( 1 + (-1.73 + 1.73i)T - 7iT^{2} \)
11 \( 1 + (3.67 + 3.67i)T + 11iT^{2} \)
13 \( 1 - 3T + 13T^{2} \)
19 \( 1 + 5.19T + 19T^{2} \)
23 \( 1 + (1.22 + 1.22i)T + 23iT^{2} \)
29 \( 1 + (-2.82 - 2.82i)T + 29iT^{2} \)
31 \( 1 + (6.92 + 6.92i)T + 31iT^{2} \)
37 \( 1 + (-5 + 5i)T - 37iT^{2} \)
41 \( 1 + (-3.53 + 3.53i)T - 41iT^{2} \)
43 \( 1 - 5.19T + 43T^{2} \)
47 \( 1 - 9.79T + 47T^{2} \)
53 \( 1 + 11.3T + 53T^{2} \)
59 \( 1 + 2.44iT - 59T^{2} \)
61 \( 1 + (1 + i)T + 61iT^{2} \)
67 \( 1 - 13.8iT - 67T^{2} \)
71 \( 1 + (-4.89 + 4.89i)T - 71iT^{2} \)
73 \( 1 + (-8 + 8i)T - 73iT^{2} \)
79 \( 1 + (1.73 - 1.73i)T - 79iT^{2} \)
83 \( 1 + 9.79iT - 83T^{2} \)
89 \( 1 + 9.89iT - 89T^{2} \)
97 \( 1 + (8 - 8i)T - 97iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.47085946814724017152374035976, −9.278978811340571339965174596914, −8.485770017091135899856449807946, −7.50766583519859629664079809361, −6.30096198134828303982483839590, −5.76162986090363613800372928503, −4.62466048342120592226687369028, −3.86170683878712086202333433152, −2.46142389505656409547699581311, −0.51172987520418644957285841301, 1.59896190946916311620561244964, 2.40374272181830390330063834005, 4.37241932155513553585420484898, 5.19179300307681077106098743386, 5.96836492550817595115657217209, 6.86940407823386106589354247471, 7.900989557291283927616306731571, 8.506572104909613047559276731704, 9.549031272983153135860356729019, 10.80882232525111372686761037672

Graph of the $Z$-function along the critical line